If you think about an infinite ruler, you can walk along it to the right (the end that goes on forever) and find every natural number X on it - if the ruler stops at any natural number X - then its a finite ruler. So an infinite ruler must have natural number X on it.Thomyum2 wrote: ↑June 28th, 2020, 12:58 pm
Yes, I follow the thought experiment, but still find it nonsensical (nothing personal - I just find most talk involving infinity to be nonsensical in general ). It cannot have 'every number' on it, because by definition, there is no such thing as 'every number'. Terms such as 'every' and 'all' only apply sensibly to finite sets or quantities.
So it also doesn't make sense to say that the infinite ruler is longer than all natural numbers. It could be said that for any given natural number X, that the infinite ruler has a length greater than X units. In other words, for any finite ruler that you could place against the infinite one, you could see that the infinite extended further. But it could not be said that length of the ruler is greater than all numbers, because that is not sensible. (And even if it was, I still don't see how that would prove that the infinite object is impossible.)
Incidentally, I have read that Cantor did make proofs that there are different kinds of infinities and that actually some infinite sets can be shown to be larger than other infinite sets - that there are actually different kinds of infinities - but that's a level of mathematics that's a bit over my head.
Now the infinite ruler goes on forever - so given any natural number X, it must go on longer than that. So an infinite ruler must be longer than all natural numbers.
I do not agree with Cantor's set theory. I find it to be very illogical. I have a lot of issues with it. Just one example:
- it claims that sets with a greater than finite number of objects actually exist in reality
- but a set is a whole number of objects only
- so the size of all sets must is constrained to be a finite number
- and also sets are formed by adding 1 object at a time, so its not possible for their size ever to become non-finite
- so (contrary to set theory) sets with a greater than finite number of objects actually DO NOT exist in reality